# Variety of pairs of product-zero matrices

Here's an old qualifying exam question I got stuck on. Consider the variety $X$ of pairs of matrices $(A,B)$ satisfying $AB = BA = 0$ (with entries in some field). What are the irreducible components of $X$? According to the question, they all have dimension $n^2$.

A related question: are the irreducible components smooth away from their loci of intersection with other components?

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I would guess that the components are something like $$X_i=\{(A,B)\vert AB=BA=0, rank A \leq i, rank B \leq n-i\},$$ but I don't really know a proof. Can anybody assist? –  Boris Datsik Aug 12 '13 at 15:17
There is a paper of Gelfand and Ponomarev Indecomposable representations of the Lorentz group'' in wich such a variety is studied over an algebraicaly closed field.